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arXiv:cond-mat/9702078 · Physical review. B, Condensed matter

Tunneling between two Luttinger liquids with long-range interaction

Maura Sassetti, Bernhard Krämer

Published Apr 15, 1997• 7 citations
arXivHTMLPDF

Abstract

The nonlinear charge transfer through a tunnel junction between two Luttinger systems is studied for repulsive, finite-range interaction between electrons on the same side, ${\mathrm{V}}_{11}$, and on different sides, ${\mathrm{V}}_{12}$, of a junction. Features of the Coulomb blockade effect are observed if ${\mathrm{V}}_{12}$=0. We predict an interaction-induced enhancement of the current if ${\mathrm{V}}_{12}$>0. When ${\mathrm{V}}_{12}$=${\mathrm{V}}_{11}$, the current is suppressed at small bias, but the ``charging energy,'' obtained from the asymptotic behavior at high-bias voltage, vanishes.

DOI
10.1103/physrevb.55.9306

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The nonlinear charge transfer through a tunnel junction between two Luttinger systems is studied for repulsive, finite-range interaction between electrons on the same side, V11{\mathrm{V}}_{11}V11​, and on different sides, V12{\mathrm{V}}_{12}V12​, of a junction. Features of the Coulomb blockade effect are observed if V12{\mathrm{V}}_{12}V12​=0. We predict an interaction-induced enhancement of the current if V12{\mathrm{V}}_{12}V12​>0. When V12{\mathrm{V}}_{12}V12​=V11{\mathrm{V}}_{11}V11​, the current is suppressed at small bias, but the ``charging energy,'' obtained from the asymptotic behavior at high-bias voltage, vanishes.
The theoretical model of the short-range interacting Luttinger liquid predicts a power-law scaling of the density of states and the momentum distribution function around the Fermi surface, which can be readily tested through tunneling experiments. However, some physical systems have long-range interaction, most notably the Coulomb interaction, leading to significantly different behaviors from the short-range interacting system. In this paper, we revisit the tunneling theory for the one-dimensional electrons interacting via the long-range Coulomb force. We show that even though in a small dynamic range of temperature and bias voltage, the tunneling conductance may appear to have a power-law decay similar to short-range interacting systems, the effective exponent is scale-dependent and slowly increases with decreasing energy. This factor may lead to the sample-to-sample variation in the measured tunneling exponents. We also discuss the crossover to a free Fermi gas at high energy and the effect of the finite size. Our work demonstrates that experimental tunneling measurements in one-dimensional electron systems should be interpreted with great caution when the system is a Coulomb Luttinger liquid.
We consider a ring of strongly interacting electrons connected to two external leads by tunnel junctions. By studying the positions of conductance resonances as a function of gate voltage and magnetic flux, the interaction parameter ggg can be determined experimentally. For a finite ring the minimum conductance is strongly influenced by device geometry and electron-electron interactions. In particular, if the tunnel junctions are close to one another, the interaction-related orthogonality catastrophe is suppressed and the valley current is unexpectedly large.
We calculate the conductances of a three-way junction of spinless Luttinger-liquid wires as functions of bias voltages applied to three independent Fermi-liquid reservoirs. In particular, we consider the setup that is characteristic of a tunneling experiment, in which the strength of electron-electron interactions in one of the arms of the junction (``tunneling tip'') is different from that in the other two arms (which together form a ``main wire''). The scaling dependence of the two independent conductances on bias voltages is determined within a fermionic renormalization-group approach in the limit of weak interactions. The solution shows that, in general, the conductances scale with the bias voltages in an essentially different way compared to their scaling with the temperature TTT. Specifically, unlike in the two-terminal setup, the nonlinear conductances cannot be generically obtained from the linear ones by simply replacing TTT with the ``corresponding'' bias voltage or the largest one. Remarkably, a finite tunneling bias voltage prevents the interaction-induced breakup of the main wire into two disconnected pieces in the limit of zero TTT and a zero source-drain voltage.
Intermode electron transitions in a two-channel Luttinger liquid (LL) are considered. At first we study chiral LL and show that for a long-range Coulomb interaction even forward-scattering intermode electron transitions are strongly renormalized by interaction at low energies. For a short-range interaction the renormalization of the electron transition rates in a chiral LL do not depend on the energy (temperature, bias voltage) involved in the tunneling process. We show, however, that the interference effects are sensitive to the strength of electron interaction, and they are significantly enhanced for strongly correlated electrons. The analyses of the intermode electron transitions is generalized for a nonchiral two-channel LL. The forward- and the backward-scattering interchannel electron transition rates are calculated and their connection to the problem of x-ray singularity in a LL is elucidated.
We study the Josephson effect through a one-dimensional system of interacting electrons, connected to two superconductors by tunnel junctions. The interactions are treated in the framework of the Luttinger model. At zero temperature, the Josephson critical current decays algebraically with increasing distance between the junctions. In a Luttinger liquid of finite size, the Josephson current depends on the total number of electrons modulo 4. These parity effects are studied for a ring, coupled capacitively to a gate voltage and threaded by a magnetic flux. Interactions introduce qualitatively new features compared to the noninteracting case.